Minimax optimality is also linked to the choice of norm, in the sense that
minimax optimality of a given procedure with respect to a norm
indicates that the procedure is optimal in the worst possible case
defined by that particular norm.
On the other hand, this does not provide much information
how the procedure performs in the worst case scenarios defined with different norms.
For example, the test proposed by [1]} is minimax optimal
with respect to the \(\ell _2\) -norm,
when the panel is spatially independent and all variances are of the same order
(in the sense described in Table REF ),
see the discussion below Theorem 2.1 of [2]}.
The test statistic proposed therein
increases the signal-to-noise ratio (while controlling for the size)
by summing up squared change point statistics;
as more cross-sectional components are considered,
the signal increases according to the squared \(\ell _2\) -norm,
i.e. linear in the number of components with a change,
whereas the noise only increases as \(\sqrt{p}\) 
due to the central limit theorem under the spatial independence assumption.
However, it does not adapt to the sparsity of change
or match the minimax optimal rate with respect to the \(\ell _\infty \) -norm;
we refer to [3]} where the authors show the phase transition boundary for the sparsity
in the minimax rate of the high-dimensional change testing problem under (REF )
(i.e. \(q_n = 1\)  at most).
